The NEAT Embedding Problem for Algebras Other than cylindric Algebras and for Infinite Dimensions

Hirsch and Hodkinson proved, for 3 ≤ m < ω and any k < ω, that the class SNrmCAm+k+1 is strictly contained in SNrmCAm+k and if k ≥ 1 then the former class cannot be defined by any finite set of first order formulas, within the latter class. We generalise this result to the following algebras of m-ary relations for which the neat reduct operator Nrm is meaningful: polyadic algebras with or without equality and substitution algebras. We also generalise this result to allow the case where m is an infinite ordinal, using quasipolyadic algebras in place of polyadic algebras (with or without equality). 1 Cylindric algebra is an algebraic correspondent of first-order logic with no constants or functions, more specifically n-dimensional cylindric algebra, CAn, is an algebraic correspondent of first-order logic restricted to n indexed variables, for finite n. An algebra in CAn is a boolean algebra together with a cylindrifier ci, which acts as a unary operator and corresponds to existential quantification of the i’th variable, and a diagonal dij element corresponding to the equality of the ith and j’th variable, for i, j < n. For m < n, the neat reduct NrmC of a C ∈ CAn is the m-dimensional cylindric algebra obtained by restricting to those elements c ∈ C such that cic = c for m ≤ i < n, and restricting to those cylindrifiers and diagonals indexed by m. If K ⊆ CAn we write NrmK for {NrmC : C ∈ K}. It is not the case that every algebra in CAm is the neat reduct of an algebra in CAn, nor need it even be a subalgebra of a neat reduct of an algebra in CAn. Furthermore, SNrmCAm+k+1 6= SNrmCAm+k, whenever 3 ≤ m < ω and k < ω [10]. A consequence of this is that there are m-variable formulas that can be proved with m+ k + 1-variables, but not with m+ k-variables, in a certain, fairly typical, proof system. Other algebras may be defined corresponding to restrictions or extensions of the n-variable first order logic described above. Because our focus is on neat reducts, we will only consider n-dimensional algebras where the cylindrifiers ci are included, or at least are definable, within the set of operators of the algebra. Without that restriction it would not be possible to define a neat reduct and our algebras would correspond to first order logic without quantifiers, we do not consider that case here. But we might choose to drop the diagonals from our signature (corresponding to first order logic without equality), or we may add c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00